[论文解读] An Illustrated Guide of the Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities

Guy Barles, Emmanuel Chasseigne|arXiv (Cornell University)|Dec 21, 2018

Stochastic processes and financial applications参考文献 91被引用 10

一句话总结

本图解指南介绍了处理具有不连续性的哈密顿-雅可比方程与控制问题的现代方法，重点在于粘性解、比较原理，以及通过分层框架处理不连续哈密顿量的方法。它统一了对余维数为1及更高维不连续性的处理方法，包括边界条件与节点问题，提供了系统化、教学化的处理方式，简化了证明并增强了概念清晰度。

ABSTRACT

This version is the last version of our book project on Hamilton-Jacobi Equations and Control Problems with discontinuities. Compared to the third version (online in december 2022), we have improved Part V (Stratified solutions for state-constraints problems) and Part VI on the applications but also the stability results for stratified solutions; we have rewritten a large part of the introduction and added guidelines for the reader. As in the previous versions, we have incorporated new results and examples, changed some points-of-view, detailed some proofs and corrected several mistakes. Version 3 had 550 pages, this one 630.As the third version, it is composed of six parts: Part I is still a toolbox with key results which are used in all the other parts. The study of the simplest case, i.e. the case of a co-dimension 1 discontinuity, is now split in two parts: in Part II, we only consider control problems and the associated Bellman Equations are treated by using only the classical notion of viscosity solutions. In this part, the methods are a combinations of control and pdes techniques. On the contrary, Part III describes purely pdes approaches which are inspired by the literature on Hamilton Jacobi Equations on networks and which can handle the case of non-convex Hamiltonians. In this part, we present two notions of solutions, namely flux-limited and junction viscosity solutions, and we study in detail their properties by providing comparison and stability results. We also show that they are ``almost'' equivalent when both make sense, i.e. for quasi-convex Hamiltonians. Part IV concerns stratified problems in $\R^N$, i.e. problems with discontinuities of any co-dimensions: the main change compared to the previous version is the introduction of a notion of ``weak'' stratified (sub)solution. In Part V, we address the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. Finally, in Part VI we describe some applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and we discuss possible extensions to problems with jumps and to ``stratified networks''.Even if we consider this version as being the final one, all comments are welcome!

研究动机与目标

• 统一并阐明处理具有不连续哈密顿量的哈密顿-雅可比方程的现代方法，特别是在控制问题背景下的应用。
• 简化并系统化处理余维数为1及以上不连续性的比较原理与粘性解技术。
• 将基于网络的方法推广至多维与分层设置，包括边界问题与状态约束问题。
• 提供一个自包含且持续演化的参考文献，通过更清晰的阐述与概念洞察，超越现有文献。
• 展示分层框架在处理复杂边界值问题（包括具有基尔霍夫型条件的问题）方面的鲁棒性。

提出的方法

• 采用“生存包”方法，帮助读者根据自身背景与目标导航本书内容，无需从头开始阅读。
• 设立专门的“基础结果”部分（第一部分），汇集可在多种框架中复用的技术工具——如比较原理与障碍构造。
• 应用消失粘性法来证明不连续设置下的解的合理性，尤其适用于通量受限解与节点解。
• 利用分层框架处理欧氏空间中任意余维数的不连续性问题，推广网络模型。
• 引入并分析伊什伊型解以处理超平面上的不连续性，解决经典粘性解方法的局限性。
• 利用利翁-苏加尼迪斯框架处理节点问题，并将其扩展至多区域与附着边界的配置。

实验结果

研究问题

• RQ1 粘性解理论如何适应处理哈密顿-雅可比方程中不连续哈密顿量的问题？
• RQ2 在超曲面或分层流形上存在不连续性时，HJB方程的比较原理在何种最小条件下成立？
• RQ3 基于网络的HJB方程方法能否在不损失一般性的前提下推广至更高维的不连续结构？
• RQ4 边界条件与传输条件如何影响具有不连续系数的HJB方程的适定性？
• RQ5 分层框架在多大程度上能统一处理节点问题、状态约束问题以及具有奇异边界条件的问题？

主要发现

• 作者通过在不同不连续类型间统一比较原理的处理方式，建立了一个系统化的粘性解理论框架。
• 本文表明，通过简化的比较论证可推导出通量受限解，优于伊梅尔特与莫内奥早期更为技术性的证明。
• 分层方法成功地将节点问题推广至多维不连续集，使得在欧氏空间中处理任意余维数的问题成为可能。
• 该框架自然地容纳边界条件与基尔霍夫型传输条件，如通过油轮问题及其相关推广所展示。
• 采用伊什伊的解概念解决了经典粘性解方法在超平面不连续性情况下的问题。
• 本书的在线持续演进格式使得证明、概念与表述能够不断优化，从而随时间推移不断提升清晰度与正确性。